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Properties

Label 12870.bf
Number of curves $2$
Conductor $12870$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 12870.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12870.bf1 12870bq2 \([1, -1, 1, -6979703, -7094584299]\) \(51583042491609575206441/9586057511268810\) \(6988235925714962490\) \([2]\) \(516096\) \(2.6181\)  
12870.bf2 12870bq1 \([1, -1, 1, -391253, -134545719]\) \(-9085904860560159241/5484993611139900\) \(-3998560342520987100\) \([2]\) \(258048\) \(2.2715\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12870.bf have rank \(0\).

Complex multiplication

The elliptic curves in class 12870.bf do not have complex multiplication.

Modular form 12870.2.a.bf

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{10} + q^{11} - q^{13} - 2 q^{14} + q^{16} + 4 q^{17} + 6 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.